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Consider the focusing semilinear wave equation in R^3 with energy-criticalnonlinearity  \partial_t^2 \psi - \Delta \psi - \psi^5 = 0, \psi(0) = \psi_0, \partial_t\psi(0) = \psi_1.  This equation admits stationary solutions of the form \phi(x, a) :=(3a)^{1/4} (1+a|x|^2)^{-1/2}, called solitons, which solve the ellipticequation -\Delta \phi - \phi^5 = 0.  Restricting ourselves to the space of symmetric solutions \psi for which\psi(x) = \psi(-x), we find a local centre-stable manifold, in a neighborhoodof \phi(x, 1), for this wave equation in the weighted Sobolev space <x>^{-1}\dot H^1 \times <x>^{-1} L^2. Solutions with initial data on the manifold existglobally in time for t \geq 0, depend continuously on initial data, preserveenergy, and can be written as the sum of a rescaled soliton and a dispersiveradiation term.  The proof is based on a new class of reverse Strichartz estimates, introducedin Beceanu-Goldberg and adapted here to the case of Hamiltonians with aresonance.

A Centre-Stable Manifold for the Energy-Critical Wave Equation in R^3 in the Symmetric Setting